Problem: Rewrite as a simplified fraction. ${2.2\overline{21} = {?}}$
Answer: Let $x$ equal the decimal. Set up two equations such that the digits after the decimal point are identical. ${\begin{align*}1000x &= 2221.2121...\\ 10x &= 22.2121...\end{align*}}$ Subtracting the two equations, we have: ${990x = 2199}$ ${x = \dfrac{2199}{990}} $ ${x = \dfrac{733}{330}} \text{or} {2 \dfrac{73}{330}}$